1,703 research outputs found
Is every toric variety an M-variety?
A complex algebraic variety X defined over the real numbers is called an
M-variety if the sum of its Betti numbers (for homology with closed supports
and coefficients in Z/2) coincides with the corresponding sum for the real part
of X. It has been known for a long time that any nonsingular complete toric
variety is an M-variety. In this paper we consider whether this remains true
for toric varieties that are singular or not complete, and we give a positive
answer when the dimension of X is less than or equal to 3.Comment: 13 page
Generalized Futaki Invariant of Almost Fano Toric Varieties, Examples
The interpretation, due to T. Mabuchi, of the classical Futaki invariant of
Fano toric manifolds is extended to the case of the Generalized Futaki
invariant, introduced by W. Ding and G. Tian, of almost Fano toric varieties.
As an application it is shown that the real part of the Generalized Futaki
invariant is positive for all degenerations of the Fano manifold V_{38},
obtained by intersection of the Veronese embedding of with codimension-two hyperplanes.Comment: 22 pages, LaTeX2
Toric Hyperkahler Varieties
Extending work of Bielawski-Dancer and Konno, we develop a theory of toric
hyperkahler varieties, which involves toric geometry, matroid theory and convex
polyhedra. The framework is a detailed study of semi-projective toric
varieties, meaning GIT quotients of affine spaces by torus actions, and
specifically, of Lawrence toric varieties, meaning GIT quotients of
even-dimensional affine spaces by symplectic torus actions. A toric hyperkahler
variety is a complete intersection in a Lawrence toric variety. Both varieties
are non-compact, and they share the same cohomology ring, namely, the
Stanley-Reisner ring of a matroid modulo a linear system of parameters.
Familiar applications of toric geometry to combinatorics, including the Hard
Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov, are
extended to the hyperkahler setting. When the matroid is graphic, our
construction gives the toric quiver varieties, in the sense of Nakajima.Comment: 32 pages, Latex; minor corrections and a reference adde
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